Arithmetic of L-Functions

For those readers who may not be familiar with it, the Park City Mathematics Institute is a program run by the Institute for Advanced Study which brings mathematicians from a variety of types of institutions, graduate students, undergraduate students, and high school teachers to the mountains of Utah for three weeks each summer to pursue their own goals and interact with each other to get a wider view of the endeavor of mathematics. The five programs they run are typically centered around a single mathematical theme: previous years’ themes have included “Moduli Spaces of Riemann Surfaces,” “Geometric Combinatorics,” “Computational Complexity Theory” and “Image Processing.” While it is not one of the official goals of the program, one of the byproducts of each year’s program has been a book collecting lecture notes from many of the graduate summer school courses as well as some of the research talks. The eighteenth volume in this series, covering the 2009 Summer Program on “Arithmetic of L-functions,” has recently been published as a co-venture of the IAS and the AMS, and for anyone wishing to learn about both the history and more recent results in the area, I would highly recommend the book.

In particular, the sections of the book include:

• An introduction by the editors, Cristian Popescu, Karl Rubin, and Alice Silverberg. This introduction gives a very brief introduction to the classical class number formula, which expresses the leading term in a power series expansion of a zeta-function of a number field K in terms of arithmetic invariants of the field, and explains how two of the central topics of the summer school, Stark’s Conjectures and the Conjecture of Birch and Swinnerton-Dyer, can be thought of as generalizations of the class number formula.
• A chapter by John Tate introducing L-functions and Stark’s Conjecture.
• A paper by Harold Stark describing the series of mathematical discoveries and papers that, over fifteen years, led to the eventual formulation of what has become known as Stark’s Conjecture in his 1980 paper in Advances of Math. This paper may not contain mathematics that is new to anyone familiar with the conjecture, but was fascinating as a story of recent mathematical history.
• A series of lectures by Cristian Popescu entitled “Integral and p-adic Refinements of the Abelian Stark Conjecture,” discussing generalizations of the Basic Stark Conjecture, including the Equivariant Tamagawa Number Conjecture, the Brumer-Stark Conjecture, including some recent results of the author and others on what he calls a Gross-type refinement of the Rubin-Stark Conjecture.
• A chapter which uses techniques from Iwasawa theory to analyze the arithmetic meaning of the values of the L-functions at negative integers, written by Manfred Kolster. These notes also discuss an analog of Stickelberger’s Theorem which discusses the annihilation of higher algebraic K-theory groups in relative abelian extensions.
• A chapter by David Burns going into more detail about the Equivariant Tamagawa Number Conjecture and its relationship to Stark’s Conjecture.
• An introduction to the main properties, theorems, and conjectures about elliptic curves, written by Alice Silverberg.
• A series of lectures by Benedict Gross, which presents the conjecture of Birch and Swinnerton-Dyer (BSD) for an elliptic curve over a global field. This conjecture essentially says that the arithmetic rank of an elliptic curve and the analytic rank of the same curve are related, and is one of the Clay Mathematics Institute Millenium Problems. These lectures introduce L-functions and some basic properties and then state the BSD conjecture in this setting and the progress that has been made towards a proof.
• Douglas Ulmer has written several lectures about elliptic curves over function fields over finite fields. These chapters begin by looking at the BSD Conjecture in this setting and its connection to Tate’s conjecture about divisors on surfaces, and then moves on to look at elliptic curves of high rank and explicit constructions in this case.
• A paper by Bryan Birch giving an account of Heegner’s proof that Gauss’ list of complex quadratic fields with class number one is complete. Heegner’s results have the reputation as being wrong because some of the details were not fully justified, but Birch argues that Heegner was “in essence entirely correct’ and aims to explain why.
• Vinayak Vatsal writes a concise introduction to the concept of complex multiplication and the elliptic curves that possess it.
• A third set of lecture notes dealing with the Equivariant Tamagawa Number Conjecture is written by Guido Kings, this time dealing with its relationship to the BSD conjecture.
• David Rohrlich has written several lectures on the use of the functional equation to determine the parity of the order of vanishing of an L-function. In particular, he discusses at length the root number and some of the subtleties that come up while computing it.
• A final series of lectures by Karl Rubin giving an introduction to the theory of Kolyvagin systems and Euler systems, based largely on the work done by the author and Barry Mazur inspired by Kolyvagin’s work bounding the size of certain Selmer groups.

I could go into more detail about the precise statements of the results in the later parts of the lecture notes, but it would be hard to say much more to a reader who does not have a passing familiarity with the topics. And I cannot imagine a better way to gain a familiarity than by reading the earlier parts of the lectures. It is worth pointing out that the lectures in this book are not in a natural linear order — as is typical with these collections, the different sets of lecture notes are aimed at slightly different levels and assume slightly different prerequisites. Some of them overlap quite a bit, and there are gaps in between some of them. Thus, this is not a textbook that one would read cover to cover, and it might best be read by jumping around somewhat, although there are occasional inconsistencies in notation and terminology.

On the flip side, the lecture notes tend to be a bit less formal and more to the point than textbooks can be, and many of the authors include exercises designed to help the reader learn and process the material. On the whole, however, I think that anyone interested in learning about the arithmetic of L-functions would be well-served by this book. Reading it might not be as good an introduction as if you had attended the PCMI Summer Schools themselves — in particular, most readers won’t have as good a view out their window — but it might be the next best thing.

Darren Glass is an Associate Professor at Gettysburg College whose research interests are primarily related to arithmetic geometry. He attended the PCMI in 1999 as a graduate student, and got a lot out of it both mathematically and otherwise. Among the many benefits was meeting the editor of MAA Reviews. He can be reached at dglass@gettysburg.edu.